In Roger Penrose's book 'The Road to Reality', while talking about complex numbers, he suggests we test that they work for "all of the necessary rules of algebra".

The rules he stated are:

$w+z=z+w$

$w+(u+z)=(w+u)+z$

$wz=zw$

$w(uz)=(wu)z$

$w(u+z)=wu+wz$

$w+0=w$

$w1=w$

$(-1)w=-w$

$(-1)(-w)=w$

That was all he listed. Are there any more "rules of algebra" in a list form such as this?

Could the following count as rules of algebra: (when /$\times x$ comes after a relation, it means times by x both sides and the like)

$\dfrac{a}{b}=c$ $/\times b$ => $a=bc$

$ab=c$ /divide b => $a=\dfrac{c}{b}$

$a=b$ $/-a$ => $b-a=0$

$b-a=0$ $/+a$ => $b=a$

But then again, perhaps this second list isn't so much a list of rules, but more a list of legal actions. However, it is these legal actions that allow one to do simple algebraic rearranging and solving. These are the four actions (multiply, divide, add, subtract) that you are allowed to do with algebraic relations.

So would my list of 4 expressions count as rules of algebra? Or would they be laws? Or just simply actions?

How far can Roger Penrose's list of rules be expanded? How many more are there of this nature?

Is there even a fundamental theory, or rule(s) of algebra? Or is the scope of algebra far too great for there to exist a list of rules and laws without it getting inconceivably messy?

4 Answers
4

I think you must be reading too deeply into what Penrose said, because I don't have any idea what would count as an answer to your question. All of those things that are listed are algebraic properties that we find useful in algebraic objects. But not all of those properties need to be shared by all interesting objects, nor is there an exhaustive list of "laws of algebra". Rings and algebras carry enough of those rules to do addition, multiplication and subtraction. A field (or division ring) carries division, also.

The last line makes it seem as if you are taking such "laws" to be members of a collection which we are gathering to describe "all of algebra" as if we were physicists seeking laws of physics. But this is not the case: in mathematics, we pick the rules and determine their consequences. Determining what rules lead to what consequences is a matter of research, but it is not expected that such discoveries lead to "the entirety" of algebra.

One more way to try convey my meaning occurred to me. The addition, subtraction and multiplication axioms for an algebra are, by convention, the shortest list of what we want an algebra to obey. They aren't to be considered as members of an incomplete and growing list of a "grand unified theory of algebra".

Those rules (called axioms) in a way are the/a minimal set that describes the numbers we are familiar with and their familiar operations (this is the rational numbers, called $\mathbb{Q}$). This set was extended step by step to "complete" it, making some further operations closed (inside the set), giving the reals $\mathbb{R}$ and complex $\mathbb{C}$ numbers, all following the above axioms.

The other rules you mention don't need to be assumed from the start (like the axioms), as they can be proved from the axioms (they are theorems). This is what mixedmath's answer explains.

As rschwieb's answer says, one of the sports mathemathicians indulge in is to invent new groud rules (or look around for cases where different sets of ground rules are apparent) and see what a minimal set of axioms would be, and where those axioms lead them.

All four of the items you proposed follow from Penrose's list. For example, you write $a = b \implies b - a = 0$. This comes from starting with $a = b$, adding the additive inverse of $a$ to get $a -a = b- a$. On the left, though, we have $a - a =0$, as that is what we mean by the additive inverse of $a$. Thus $b - a = 0$.

Similarly, your other rules follow from his. His list is complete, but there are multiple 'things' that follow those rules. The set of real numbers, of the set of fractions, or the complex numbers, for instance.

An algebra is a the overarching term for a space, where the elements “interact” with each other according to certain operations/dynamics. What these operations may be, and what properties they may have, lead to the “name-calling” — i. e. naming the algebra as a group, or a ring, or field, or a boolean algebra, or a C* or $\ldots$ etc.